In quantum mechanics, an energy level is said to be degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independenteigenstate with the same eigenvalue. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

Degeneracy plays a fundamental role in quantum statistical mechanics. For a N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.

Mathematically, the possible states of a quantum mechanical system may be treated as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If A is a N × N matrix, X a non-zero vector, and λ be a scalar, such that , then the scalar λ is said to be an eigenvalue of A and the vector X is said to be the eigenvector corresponding to λ. Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue λ form a subspace of Cn, which is called the eigenspace of λ. An eigenvalue λ which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e., and , where and are linearly independent eigenvectors.The dimensionality of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional. The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue.

Proof of the above theorem.

If represents the Hamiltonian operator and and are two eigenstates corresponding to the same eigenvalue E, then

Let , where and are complex(in general) constants, be any linear combination of and . Then,

In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. However, if the Hamiltonian has a degenerate eigenvalue of degree gn, the eigenstates associated with it form a vector subspace of dimension gn. In such a case, several final states can be possibly associated with the same result , all of which are linear combinations of the gnorthonormal eigenvectors .

In this case, the probability that the energy value measured for a system in the state will yield the value is given by the sum of the probabilities of finding the system in each of the states in this basis, i.e.

This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual understanding of more complex systems.

In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function moving in a one-dimensional potential , the time-independent Schrödinger equation can be written as

Since the number of independent eigenfunctions for a given energy is maximally equal to two, the degree of degeneracy never exceeds two. It can be proved that in one dimension, there are no degenerate bound states for normalizable wave functions. This is true provided the potential under consideration is bounded from below and piecewise continuous.

Proof of the above theorem.

Considering a one-dimensional quantum system in a potential with degenerate states and corresponding to the same energy eigenvalue , writing the time-independent Schrödinger equation for the system:

Multiplying the first equation by and the second by and subtracting one from the other, we get:

Integrating both sides

In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: where is, in general, a complex constant. So, the two degenerate wave functions are not linearly independent. For bound state eigenfunctions, the wave function approaches zero in the limit or , so that the above constant is one and we have no degeneracy.

Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made of monoatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems.

Degrees of degeneracy of different energy levels for a particle in a square box

In this case, the dimensions of the box and the energy eigenvalues are given by

Since and can be interchanged without changing the energy, each energy level is at least twice as degenerate when and are different. Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same.

If two operators and commute, i.e. , then for every eigenvector of , is also an eigenvector of with the same eigenvalue. However, if this eigenvalue, say , is degenerate, it can be said that belongs to the eigenspace of , which is said to be globally invariant under the action of .

For two commuting observables A and B, one can construct an orthonormal basis of the state space with eigenvectors common to the two operators. However, is a degenerate eigenvalue of , then it is an eigensubspace of that is invariant under the action of , so the representation of in the eigenbasis of is not a diagonal but a block diagonal matrix, i.e. the degenerate eigenvectors of are not, in general, eigenvectors of . However, it is always possible to choose, in every degenerate eigensubspace of , a basis of eigenvectors common to and .

If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. On the other hand, if one or several eigenvalues of are degenerate, specifying an eigenvalue is not sufficient to characterize a basis vector. If, by choosing an observable , which commutes with , it is possible to construct an orthonormal basis of eigenvectors common to and , which is unique, for each of the possible pairs of eigenvalues {a,b}, then and are said to form a complete set of commuting observables. However, if a unique set of eigenvectors can still not be specified, for at least one of the pairs of eigenvalues, a third observable , which commutes with both and can be found such that the three form a complete set of commuting observables.

It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator that commutes with the Hamiltonian. These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system.

The parity operator is defined by its action in the representation of changing r to -r, i.e.

The eigenvalues of P can be shown to be limited to , which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while that with eigenvalue −1 is said to be odd.

Now, an even operator is one that satisfies,

while an odd operator is one that satisfies

Since is even, if the potential V(r) is even, the Hamiltonian is said to be an even operator. In that case, if each of its eigenvalues are non-degenerate, each eigenvector is necessarily an eigenstate of P, and therefore it is possible to look for the eigenstates of among even and odd states. However, if one of the energy eigenstates has no definite parity, it can be asserted that the corresponding eigenvalue is degenerate, and is an eigenvector of with the same eigenvalue as .

The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find the energy levels and degeneracies without solving the Schrödinger equation.

Proof that the existence of an operator which commutes with the Hamiltonian results in degeneracy.

Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Let us consider a symmetry operation associated with a unitary operatorS. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that , since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, we have

Now, if is an energy eigenstate,

where E is the corresponding energy eigenvalue.

which means that is also an energy eigenstate with the same eigenvalue E, i.e. the two states and are degenerate.

In cases where S is characterized by a continuous parameter, all states of the form have the same energy eigenvalue.

The set of all operators which commute with the Hamiltonian of a quantum system are said to form the Symmetry group of the Hamiltonian. The commutators of the generators of this group[disambiguation needed] determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.

This is also called a geometrical or normal degeneracy and arises due to the presence of some kind of symmetry in the system under consideration, i.e. the invariance of the Hamiltonian under a certain operation, as described above. The representation obtained from a normal degeneracy is irreducible and the corresponding eigenfunctions form a basis for this representation.

It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in the system. It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete. These degeneracies are connected to the existence of bound orbits in classical Physics.

For a particle is a central 1/r potential, the Laplace–Runge–Lenz vector is a conserved quantity resulting from an accidental degeneracy, in additional to the conservation of angular momentum due to rotational invariance.

For a particle moving on a cone under the influence of 1/r and r2 potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of an equivalent of the Runge-Lenz vector, in addition to one component of the angular momentum vector. These quantities generate SU(2) symmetry for both potentials.

A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry. The symmetry multiplets in this case are the Landau levels which are infinitely degenerate.

In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total orbital angular momentum, its component along the z-direction, , total spin angular momentum and its z-component , the nuclear spin angular momentum and . Since the energy levels in the hydrogen atom depend only on the principal quantum numbern, all the n states corresponding to → have the same energy, and similarly,the states with → have the same energy, and are degenerate states. This is an essential degeneracy, which arises from the factor that the quantum number does not appear in the radial part of the time-independent Schrödinger equation. The degree of degeneracy of the energy level En is therefore n2, which is doubled to the possibility of two spin states for an electron.

The wavefunctions of the Hydrogen atom corresponding to different values of the azimuthal quantum number l may also have the same energy, and the resulting degeneracies are accidental.

Since the state space of such a particle is the tensor product of the state spaces associated with the individual one-dimensional wave functions, the time-independent Schrödinger equation for such a system is given by-

So, the energy eigenvalues are

or,

where n is a non-negative integer. So, the energy levels are degenerate and the degree of degeneracy is equal to the number of different sets satisfying

The degeneracy in a quantum mechanical system may be removed if the underlying symmetry is broken by an external perturbation. This causes splitting in the degenerate energy levels. This is essentially a splitting of the original irreducible representations into lower-dimensional such representations of the perturbed system.

Mathematically, the splitting due to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation scheme that can be applied to find the solution to the eigenvalue equation for the Hamiltonian H of a quantum system with an applied perturbation, given the solution for the Hamiltonian H0 for the unperturbed system. It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the perturbed system near them. The correct basis to choose is one that diagonalizes the perturbation Hamiltonian within the degenerate subspace.

Lifting of degeneracy by first-order degenerate perturbation theory.

Consider an unperturbed Hamiltonian and perturbation , so that the perturbed Hamiltonian

The perturbed eigenstate, for no degeneracy, is given by-

The perturbed energy eigenket as well as higher order energy shifts diverge when , i.e., in the presence of degeneracy in energy levels. Assuming possesses N degenerate eigenstates with the same energy eigenvalue E, and also in general some non-degenerate eigenstates. A perturbed eigenstate can be written as a linear expansion in the unperturbed degenerate eigenstates as-

where refer to the perturbed energy eigenvalues. Since is a degenerate eigenvalue of ,

Premultiplying by another unperturbed degenerate eigenket gives-

This is an eigenvalue problem, and writing , we have-

The N eigenvalues obtained by solving this equation give the shifts in the degenerate energy level due to the applied perturbation, while the eigenvectors give the perturbed states in the unperturbed degenerate basis . To choose the good eigenstates from the beginning, it is useful to find an operator which commutes with the original Hamiltonian and has simultaneous eigenstates with it.

A two-level system essential refers to a physical system having two states whose energies are close together and very different from those of the other states of the system. All calculations for such a system are performed on a two-dimensional subspace of the state space.

If the ground state of a physical system is two-fold degenerate, any coupling between the two corresponding states lowers the energy of the ground state of the system, and makes it more stable.

If and are the energy levels of the system, such that , and the perturbation is represented in the two-dimensional subspace as the following 2X2 matrix

then the perturbed energies are

Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent property of the system include-

Benzene, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.

Ammonia molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.

H2+ molecule, in which the electron may be localized around either of the two nuclei.

The corrections to the Coulomb interaction between the electron and the proton in a Hydrogen atom due to relativistic motion and spin-orbit coupling result in breaking the degeneracy in energy levels for different values of l corresponding to a single principal quantum number n.

The perturbation Hamiltonian due to relativistic correction is given by

where is the momentum operator and is the mass of the electron. The first-order relativistic energy correction in the basis is given by

The spin-orbit interaction refers to the interaction between the intrinsic magnetic moment of the electron with the magnetic field experienced by it due to the relative motion with the proton. The interaction Hamiltonian is

which may be written as

The first order energy correction in the basis where the perturbation Hamiltonian is diagonal, is given by

where is the Bohr radius. The total fine-structure energy shift is given by

The splitting of the energy levels of an atom when placed in an external magnetic field because of the interaction of the magnetic moment of the atom with the applied field is known as the Zeeman effect.

Taking into consideration the orbital and spin angular momenta, and , respectively, of a single electron in the Hydrogen atom, the perturbation Hamiltonian is given by-

where and . Thus,

Now, in case of the weak-field Zeeman effect, when the applied field is weak compared to the internal field, the spin-orbit coupling dominates and and are not separately conserved. The good quantum numbers are n, l, j and mj, and in this basis, the first order energy correction can be shown to be given by

, where

is called the Bohr Magneton.Thus, depending on the value of , each degenerate energy level splits into several levels.

Lifting of degeneracy by an external magnetic field

In case of the strong-field Zeeman effect, when the applied field is strong enough, so that the orbital and spin angular momenta decouple, the good quantum numbers are now n,l,ml and ms. Here, Lz and Sz are conserved, so the perturbation Hamiltonian is given by-

The splitting of the energy levels in a Hydrogen atom when subjected to an external electric field is known as the Stark effect.

Here, the perturbation Hamiltonian is-

if the electric field is chosen along the z-direction.

The energy corrections due to the applied field are given by the expectation value of is the basis. It can be shown by the selection rules that when and .

The degeneracy is lifted only for certain states obeying the selection rules, in the first order. The first-order splitting in the energy levels for the degenerate states and , both corresponding to n=2, is given by .